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From: Dave Matthews
Subject: Re: Knot or not?
Date: 31 Oct 2004 14:51:03
Message: <41854227@news.povray.org>
Andrew the Orchid wrote:


> Well, one of us has fluffed up somewhere :-S Since the procedure you 
> undertook is simpler, I suspect it was me :-$
> 
> Andrew.

Well, again studiously avoiding doing anything resembling mathematics 
myself, I downloaded a "KnotTheory" Mathematica package ( 
http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html ), and 
typed in Jones[TorusKnot[3,2]][q], which produced q + q^3 - q^4 (which 
is the same as for one of the Trefoil knots, and the picture the package 
gives for the (3,2) Torus knot looks like a trefoil.  See the 
attachment.  If I recall correctly (and that's a big ^if^), the 
exponents in the Jones polynomial should always work out to be positive 
or negative integers.  I suppose now I'm going to have to sit down and 
think it through again.  I'm too old for this ;-)

Dave Matthews


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From: Dave Matthews
Subject: Re: Knot or not?
Date: 31 Oct 2004 15:08:41
Message: <41854649$1@news.povray.org>
Dave Matthews wrote:

> Andrew the Orchid wrote:
> 
> 
>> Well, one of us has fluffed up somewhere :-S Since the procedure you 
>> undertook is simpler, I suspect it was me :-$
>>
>> Andrew.
> 
> 
> Well, again studiously avoiding doing anything resembling mathematics 
> myself, I downloaded a "KnotTheory" Mathematica package ( 
> http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html ), and 
> typed in Jones[TorusKnot[3,2]][q], which produced q + q^3 - q^4 (which 
> is the same as for one of the Trefoil knots, and the picture the package 
> gives for the (3,2) Torus knot looks like a trefoil.  See the 
> attachment.  If I recall correctly (and that's a big ^if^), the 
> exponents in the Jones polynomial should always work out to be positive 
> or negative integers.  I suppose now I'm going to have to sit down and 
> think it through again.  I'm too old for this ;-)
> 
> Dave Matthews
> 
> ------------------------------------------------------------------------
> 
Now I notice that the handed-ness is reversed (he said, replying to 
himself.)  I see why:  I actually had to scale my earlier image by -1 to 
get Mike Williams' knot, and then also scaled the trefoil.  So I guess 
we should get q^-1 + q^-3 - q^-4 for the Jones polynomial of either or both.

Dave Matthews


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From: Andrew the Orchid
Subject: Re: Knot or not?
Date: 31 Oct 2004 15:34:01
Message: <41854c39@news.povray.org>
> Well, again studiously avoiding doing anything resembling mathematics 
> myself, I downloaded a "KnotTheory" Mathematica package ( 
> http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html ), and 
> typed in Jones[TorusKnot[3,2]][q], which produced q + q^3 - q^4 (which 
> is the same as for one of the Trefoil knots, and the picture the package 
> gives for the (3,2) Torus knot looks like a trefoil.  See the 
> attachment.  If I recall correctly (and that's a big ^if^), the 
> exponents in the Jones polynomial should always work out to be positive 
> or negative integers.  I suppose now I'm going to have to sit down and 
> think it through again.  I'm too old for this ;-)

*You* have Mathematica? :-|

[Is envious]


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From: Dave Matthews
Subject: Re: Knot or not?
Date: 1 Nov 2004 08:45:33
Message: <41863dfd@news.povray.org>
Andrew the Orchid wrote:

> *You* have Mathematica? :-|
> 
> [Is envious]

My employers have Mathematica (I teach at a community college.)  Along 
with lots of vacation time, it's one of the perks.

Dave Matthews


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From: Andrew the Orchid
Subject: Re: Knot or not?
Date: 1 Nov 2004 15:34:37
Message: <41869ddd$1@news.povray.org>
>> *You* have Mathematica? :-|
>>
>> [Is envious]
> 
> 
> My employers have Mathematica (I teach at a community college.)  Along 
> with lots of vacation time, it's one of the perks.

OK.

OTOH, it's a college...

But still, one of my long-held dreams is to one day own Mathematica. 
(Although just being able to use it would also be kinda cool. Heh!)

If only it didn't cost almost as much as my car... (Well, if you're a 
student [studying in the right place], it doesn't cost that much. *sigh*)

Andrew.


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From: Andrew the Orchid
Subject: Re: Knot or not?
Date: 1 Nov 2004 15:49:57
Message: <4186a175@news.povray.org>
> Well, if it's the same as the first knot, below, and I think it is, then 
> it's a "3,2 Torus Knot" (which I generated using KnotPlot -- 
> http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html )
> 
> In which case, using KnotPlot, I deformed it (continuously, so it should 
> be topologically equivalent) to the second knot, below, which looks 
> somewhat familiar ;-)

Never heard of KnotPlot before.

However, in a stunning coincidence, today I did a search on Google 
[relating to knot theory], and I unded up reading a paper on drawing 
knots. And about half way through the introduction, I find that this 
paper was written by the author of something called KnotPlot...

...so I guess by the time I finish all 250+ pages, I'll know all about 
it! :-D

Andrew.


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